I · Sets & Venn Diagrams
Q 01
If A = {1, 3, 5, 7, 9} and B = {3, 6, 9, 12}, what is A ∩ B?
⚡ Memory Key
∩ = AND = IN BOTH — Intersection means elements that appear in both sets simultaneously. Think: "n" in ∩ = iN both.
Q 02
In a class of 30 students, 18 play soccer and 14 play basketball. If 5 play both, how many play at least one sport?
|A ∪ B| = |A| + |B| − |A ∩ B|
⚡ Memory Key
UNION FORMULA: ADD then SUBTRACT overlap — You double-count the middle, so subtract it once. "Plus minus the middle."
Q 03
⚠ Common mistake — read carefully!
Let U = {1,2,3,4,5,6,7,8}, A = {2,4,6,8}. What is A' (complement of A)?
Let U = {1,2,3,4,5,6,7,8}, A = {2,4,6,8}. What is A' (complement of A)?
⚡ Memory Key
COMPLEMENT = NOT in A — A' is everything in the Universal set U that is not in A. Always check what U is first!
Q 04
In a Venn diagram with sets A and B inside universal set U, n(U)=40, n(A)=16, n(B)=20, n(A∩B)=7. How many elements are in neither A nor B?
⚡ Memory Key
NEITHER = U − Union — Step 1: find n(A∪B) using the formula. Step 2: subtract from n(U).
Q 05
⚠ Tricky notation alert!
Which statement is always TRUE?
Which statement is always TRUE?
⚡ Memory Key
∅ ⊆ EVERY set — The empty set is a subset of all sets. And A ⊆ A always (reflexive). But A ⊂ A is FALSE (proper subset needs A ≠ B).
II · Probability
Q 06
A bag contains 4 red, 3 blue, and 5 green marbles. One marble is picked at random. What is P(red)?
⚡ Memory Key
P = FAVORABLE / TOTAL — Always count all outcomes first (denominator), then count only what you want (numerator).
Q 07
A fair coin is flipped and a fair die is rolled. What is the probability of getting Heads AND a 4?
⚡ Memory Key
INDEPENDENT → MULTIPLY — When events don't affect each other: P(A AND B) = P(A) × P(B). "AND = ×"
Q 08
⚠ Most students over-calculate this!
The probability it rains tomorrow is 0.35. What is the probability it does NOT rain?
The probability it rains tomorrow is 0.35. What is the probability it does NOT rain?
⚡ Memory Key
P(A') = 1 − P(A) — All probabilities sum to 1. Complement rule: flip it! "One minus the given."
Q 09
⚠ Classic exam trap — the condition changes everything!
A box has 3 red and 2 blue balls. Two balls are drawn without replacement. What is P(both red)? P(A then B) = P(A) × P(B | A was already taken)
A box has 3 red and 2 blue balls. Two balls are drawn without replacement. What is P(both red)? P(A then B) = P(A) × P(B | A was already taken)
⚡ Memory Key
WITHOUT REPLACEMENT → denominator drops! — After 1st draw: total goes from 5→4, red goes from 3→2. Always reduce both numbers.
Q 10
Events A and B are mutually exclusive. P(A) = 0.3, P(B) = 0.4. Find P(A ∪ B).
⚡ Memory Key
MUTUALLY EXCLUSIVE → P(A∩B) = 0 — They can't happen together, so intersection is zero. Union formula simplifies to just P(A) + P(B).
Q 11
A student passes Math with probability 0.7 and Science with probability 0.6 (independent). What is the probability of passing at least one?
⚡ Memory Key
"AT LEAST ONE" → USE COMPLEMENT — P(at least 1) = 1 − P(none). P(fail both) = 0.3 × 0.4. Much faster than listing all cases!
Q 12
⚠ Formula-dependent — must know this!
P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.2. Find P(A | B). P(A | B) = P(A ∩ B) / P(B)
P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.2. Find P(A | B). P(A | B) = P(A ∩ B) / P(B)
⚡ Memory Key
CONDITIONAL = INTERSECTION ÷ GIVEN — "Given B" means B is your new sample space. Divide by what you're given!
III · Counting & Combinations
Q 13
Evaluate 5!
⚡ Memory Key
n! = n × (n−1) × ... × 2 × 1 — "Count down, multiply all the way." Also: 0! = 1 (memorize this!)
Q 14
In how many ways can 4 students be arranged in a line from a group of 7?
ⁿPᵣ = n! / (n−r)! → ⁷P₄ = ?
⚡ Memory Key
PERMUTATION = ORDER MATTERS — "Arrange" / "in a line" / "first/second/third" → use P. Think: "Placing people in Positions."
Q 15
A committee of 3 people is chosen from 8 candidates. How many different committees are possible?
ⁿCᵣ = n! / (r! × (n−r)!) → ⁸C₃ = ?
⚡ Memory Key
COMBINATION = ORDER DOESN'T MATTER — "Choose" / "select" / "committee" → use C. Think: "C for Choose, order doesn't Count."
Q 16
⚠ The most common mistake in counting!
A password uses 3 different digits chosen from {0,1,2,3,4,5,6,7,8,9}. The order of digits matters. How many passwords are possible?
A password uses 3 different digits chosen from {0,1,2,3,4,5,6,7,8,9}. The order of digits matters. How many passwords are possible?
⚡ Memory Key
ASK: Does order matter? — Password 123 ≠ 321 → ORDER MATTERS → use Permutation. Choosing a team: {A,B,C} = {C,A,B} → ORDER DOESN'T MATTER → use Combination.
Q 17
A restaurant offers 3 starters, 5 mains, and 2 desserts. How many different 3-course meals are possible?
⚡ Memory Key
INDEPENDENT CHOICES → MULTIPLY — "And then" = multiply. "Or" = add. Stages that happen one after another: always multiply.
Q 18
A hand of 5 cards is dealt from a standard deck of 52 cards. Which expression gives the number of possible hands?
⚡ Memory Key
CARDS / LOTTERY = COMBINATION — You hold {A,K,Q,J,10} — the order you were dealt doesn't matter. Always C for cards!
Q 19
A bag has 6 red and 4 blue balls. Two balls are drawn at random. What is the probability both are red? (Use combinations.)
P(both red) = ⁶C₂ / ¹⁰C₂
⚡ Memory Key
P = (FAVORABLE combos) / (TOTAL combos) — When drawing without caring about order: use C in numerator AND denominator. Same structure as basic probability!
Q 20
🏆 BOSS QUESTION — combines all three topics!
In a group of 10 students, 6 study Math and 4 study Art (no overlap). A committee of 3 is randomly selected. What is the probability the committee has exactly 2 Math students and 1 Art student? P = (⁶C₂ × ⁴C₁) / ¹⁰C₃
In a group of 10 students, 6 study Math and 4 study Art (no overlap). A committee of 3 is randomly selected. What is the probability the committee has exactly 2 Math students and 1 Art student? P = (⁶C₂ × ⁴C₁) / ¹⁰C₃
⚡ Memory Key
SPLIT & MULTIPLY, then DIVIDE by TOTAL — Choose from each group separately (multiply), then divide by all possible committees. "Groups × Groups ÷ Total."